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Recently I was playing around with the idea of graphs of projectile motion and I started wondering what it would look like if someone were to graph every parabolic trajectory of a single initial velocity over all angles between o and 90 degrees above the horizontal. I decided to find out so in excel, I used the equation y=tanθ⋅x−x^{2}⋅g/(2⋅v_{i}^{2}⋅cos^{2}θ), with an initial velocity of 6, a gravitational acceleration of 9.8, and 180 different angles, all between 0 and 90 degrees. The resulting chart this:
What I want to know is if it is possible to derive an equation for the curve above the area that this chart is approximating, partly out of curiosity about the equation itself and partly out of curiosity about whether it would be possible to then integrate that equation to find out the area of under that curve. I am also wondering if that shape has a name.
In addition, while playing around with this idea, I also made this chart:
which graphs only the vertices of 360 versions of the above equation. Each vertex was obtained using x=(v_{i}^{2}⋅sin(2θ))/(2⋅g) and y=(v_{i}^{2}⋅sin^{2}θ)/(2⋅g). Like with the other chart, I'm wondering if it is possible to derive an equation for the graph that this chart is approximating.
What I want to know is if it is possible to derive an equation for the curve above the area that this chart is approximating, partly out of curiosity about the equation itself and partly out of curiosity about whether it would be possible to then integrate that equation to find out the area of under that curve. I am also wondering if that shape has a name.
In addition, while playing around with this idea, I also made this chart:
which graphs only the vertices of 360 versions of the above equation. Each vertex was obtained using x=(v_{i}^{2}⋅sin(2θ))/(2⋅g) and y=(v_{i}^{2}⋅sin^{2}θ)/(2⋅g). Like with the other chart, I'm wondering if it is possible to derive an equation for the graph that this chart is approximating.
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